Determining the Domain of the Graphed Function

Oct 13, 2025 by recoveryshake.com 49 views

The coordinate grid is a fundamental tool in mathematics, used to visualize and analyze functions. In this article, we will explore the concept of function translation and its application to the graph of $y=\sqrt[3]{x-1}+3$ . We will also determine the domain of the graphed function.

What is Function Translation?

Function translation is a process of shifting or transforming a function to create a new function. This can be done horizontally, vertically, or both. In the case of the graph of $y=\sqrt[3]{x-1}+3$ , we can see that it is a translation of the function $y=\sqrt[3]{x}$ .

Understanding the Graph of $y=\sqrt[3]{x-1}+3$

The graph of $y=\sqrt[3]{x-1}+3$ is a translation of the graph of $y=\sqrt[3]{x}$ . To understand this, let's break down the function. The cube root function, $\sqrt[3]{x}$ , is a function that takes a value $x$ and returns its cube root. When we add 1 to the input $x$ , we are shifting the graph of $y=\sqrt[3]{x}$ one unit to the right. This is because the input $x$ is now $x-1$ , which is one unit less than the original input.

Determining the Domain of the Graphed Function

The domain of a function is the set of all possible input values for which the function is defined. In the case of the graph of $y=\sqrt[3]{x-1}+3$ , we need to determine the domain of the function $y=\sqrt[3]{x-1}$ . Since the cube root function is defined for all real numbers, the domain of the function $y=\sqrt[3]{x-1}$ is all real numbers.

However, we need to consider the translation of the graph. When we shift the graph one unit to the right, the domain of the function changes. The new domain is all real numbers greater than 1, since the input $x$ is now $x-1$ , which is one unit less than the original input.

Conclusion

In conclusion, the graph of $y=\sqrt[3]{x-1}+3$ is a translation of the function $y=\sqrt[3]{x}$ . The domain of the graphed function is all real numbers greater than 1, which can be represented as $\{x \mid 1\ \textless \ x\ \textless \ 5\}$ .

Answer

The correct answer is A. $\{x \mid 1\ \textless \ x\ \textless \ 5\}$ .

Additional Information

It's worth noting that the domain of the graphed function is not limited to the interval $[1, 5]$ . The function is defined for all real numbers greater than 1, which means that the domain is an open interval.

Final Thoughts

In this article, we explored the concept of function translation and its application to the graph of $y=\sqrt[3]{x-1}+3$ . We determined the domain of the graphed function and concluded that it is all real numbers greater than 1. This article provides a comprehensive understanding of function translation and its importance in mathematics.

References

[1] "Function Translation" by Math Open Reference. Retrieved from https://www.mathopenref.com/functranslation.html
[2] "Cube Root Function" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/CubeRootFunction.html

Keywords

Function Translation
Cube Root Function
Domain of a Function
Coordinate Grid
Graph of a Function
Q&A: Understanding Function Translation and the Coordinate Grid ================================================================

In our previous article, we explored the concept of function translation and its application to the graph of $y=\sqrt[3]{x-1}+3$ . We determined the domain of the graphed function and concluded that it is all real numbers greater than 1. In this article, we will answer some frequently asked questions about function translation and the coordinate grid.

Q: What is function translation?

A: Function translation is a process of shifting or transforming a function to create a new function. This can be done horizontally, vertically, or both.

Q: What is the difference between horizontal and vertical translation?

A: Horizontal translation involves shifting the graph of a function to the left or right, while vertical translation involves shifting the graph of a function up or down.

Q: How do I determine the domain of a graphed function?

A: To determine the domain of a graphed function, you need to consider the restrictions on the input values. For example, if a function has a square root, the input values must be non-negative.

Q: What is the significance of the coordinate grid in mathematics?

A: The coordinate grid is a fundamental tool in mathematics, used to visualize and analyze functions. It allows us to represent functions as graphs and to identify patterns and relationships between variables.

Q: How do I graph a function on the coordinate grid?

A: To graph a function on the coordinate grid, you need to identify the x and y intercepts of the function. You can then use these intercepts to draw the graph of the function.

Q: What is the difference between a function and a relation?

A: A function is a relation between a set of inputs (called the domain) and a set of outputs (called the range). A relation, on the other hand, is a set of ordered pairs that do not necessarily satisfy the properties of a function.

Q: How do I determine if a relation is a function?

A: To determine if a relation is a function, you need to check if each input value corresponds to exactly one output value. If this is the case, then the relation is a function.

Q: What is the significance of the cube root function in mathematics?

A: The cube root function is a fundamental function in mathematics, used to find the cube root of a number. It is defined as the inverse of the cube function.

Q: How do I graph the cube root function on the coordinate grid?

A: To graph the cube root function on the coordinate grid, you need to identify the x and y intercepts of the function. You can then use these intercepts to draw the graph of the function.

Q: What are some common applications of function translation in real-world problems?

A: Function translation has many applications in real-world problems, such as:

Modeling population growth and decline
Analyzing financial data and predicting stock prices
Understanding the behavior of physical systems, such as the motion of objects
Developing algorithms for computer science and engineering applications

Conclusion

In conclusion, function translation is a fundamental concept in mathematics, used to analyze and understand functions. The coordinate grid is a powerful tool for visualizing and analyzing functions, and it has many applications in real-world problems. By understanding function translation and the coordinate grid, you can develop a deeper appreciation for the beauty and power of mathematics.

References

[1] "Function Translation" by Math Open Reference. Retrieved from https://www.mathopenref.com/functranslation.html
[2] "Cube Root Function" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/CubeRootFunction.html
[3] "Coordinate Grid" by Khan Academy. Retrieved from https://www.khanacademy.org/math/geometry/coordinate-geometry

Keywords

Function Translation
Cube Root Function
Coordinate Grid
Domain of a Function
Range of a Function
Graph of a Function
Inverse Functions
Mathematical Modeling